Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  signslema Structured version   Unicode version

Theorem signslema 26978
Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)
Hypotheses
Ref Expression
signslema.1  |-  ( ph  ->  E  e.  NN0 )
signslema.2  |-  ( ph  ->  F  e.  NN0 )
signslema.3  |-  ( ph  ->  G  e.  NN0 )
signslema.4  |-  ( ph  ->  H  e.  NN0 )
signslema.5  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
signslema.6  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
Assertion
Ref Expression
signslema  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )

Proof of Theorem signslema
StepHypRef Expression
1 signslema.5 . . . . . 6  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
21simpld 459 . . . . 5  |-  ( ph  ->  E  <  G )
32adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  E  <  G )
4 signslema.4 . . . . . . . . . 10  |-  ( ph  ->  H  e.  NN0 )
54nn0cnd 10653 . . . . . . . . 9  |-  ( ph  ->  H  e.  CC )
6 signslema.2 . . . . . . . . . 10  |-  ( ph  ->  F  e.  NN0 )
76nn0cnd 10653 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
85, 7subcld 9734 . . . . . . . 8  |-  ( ph  ->  ( H  -  F
)  e.  CC )
9 signslema.3 . . . . . . . . . 10  |-  ( ph  ->  G  e.  NN0 )
109nn0cnd 10653 . . . . . . . . 9  |-  ( ph  ->  G  e.  CC )
11 signslema.1 . . . . . . . . . 10  |-  ( ph  ->  E  e.  NN0 )
1211nn0cnd 10653 . . . . . . . . 9  |-  ( ph  ->  E  e.  CC )
1310, 12subcld 9734 . . . . . . . 8  |-  ( ph  ->  ( G  -  E
)  e.  CC )
148, 13subeq0ad 9744 . . . . . . 7  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  <-> 
( H  -  F
)  =  ( G  -  E ) ) )
1514biimpa 484 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( H  -  F )  =  ( G  -  E ) )
1615breq2d 4319 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
0  <  ( H  -  F )  <->  0  <  ( G  -  E ) ) )
176nn0red 10652 . . . . . . 7  |-  ( ph  ->  F  e.  RR )
184nn0red 10652 . . . . . . 7  |-  ( ph  ->  H  e.  RR )
1917, 18posdifd 9941 . . . . . 6  |-  ( ph  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2019adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2111nn0red 10652 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
229nn0red 10652 . . . . . . 7  |-  ( ph  ->  G  e.  RR )
2321, 22posdifd 9941 . . . . . 6  |-  ( ph  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2516, 20, 243bitr4rd 286 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  F  <  H ) )
263, 25mpbid 210 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  F  <  H )
27 0re 9401 . . . . . 6  |-  0  e.  RR
2827a1i 11 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  e.  RR )
2922, 21resubcld 9791 . . . . . 6  |-  ( ph  ->  ( G  -  E
)  e.  RR )
3029adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  e.  RR )
3118, 17resubcld 9791 . . . . . 6  |-  ( ph  ->  ( H  -  F
)  e.  RR )
3231adantr 465 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( H  -  F )  e.  RR )
332adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  E  <  G )
3423adantr 465 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
3533, 34mpbid 210 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( G  -  E
) )
36 2pos 10428 . . . . . . . 8  |-  0  <  2
37 breq2 4311 . . . . . . . 8  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  (
0  <  ( ( H  -  F )  -  ( G  -  E ) )  <->  0  <  2 ) )
3836, 37mpbiri 233 . . . . . . 7  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  0  <  ( ( H  -  F )  -  ( G  -  E )
) )
3938adantl 466 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( ( H  -  F )  -  ( G  -  E )
) )
4029, 31posdifd 9941 . . . . . . 7  |-  ( ph  ->  ( ( G  -  E )  <  ( H  -  F )  <->  0  <  ( ( H  -  F )  -  ( G  -  E
) ) ) )
4140biimpar 485 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( H  -  F
)  -  ( G  -  E ) ) )  ->  ( G  -  E )  <  ( H  -  F )
)
4239, 41syldan 470 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  <  ( H  -  F
) )
4328, 30, 32, 35, 42lttrd 9547 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( H  -  F
) )
4419adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
4543, 44mpbird 232 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  F  <  H )
465, 10, 7, 12sub4d 9783 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  =  ( ( H  -  F )  -  ( G  -  E ) ) )
47 signslema.6 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
4846, 47eqeltrrd 2518 . . . 4  |-  ( ph  ->  ( ( H  -  F )  -  ( G  -  E )
)  e.  { 0 ,  2 } )
49 ovex 6131 . . . . 5  |-  ( ( H  -  F )  -  ( G  -  E ) )  e. 
_V
5049elpr 3910 . . . 4  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  e.  { 0 ,  2 }  <->  ( (
( H  -  F
)  -  ( G  -  E ) )  =  0  \/  (
( H  -  F
)  -  ( G  -  E ) )  =  2 ) )
5148, 50sylib 196 . . 3  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  \/  ( ( H  -  F )  -  ( G  -  E
) )  =  2 ) )
5226, 45, 51mpjaodan 784 . 2  |-  ( ph  ->  F  <  H )
531simprd 463 . . . . 5  |-  ( ph  ->  -.  2  ||  ( G  -  E )
)
5453adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( G  -  E ) )
5515breq2d 4319 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
2  ||  ( H  -  F )  <->  2  ||  ( G  -  E
) ) )
5654, 55mtbird 301 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( H  -  F ) )
57 2z 10693 . . . . . . 7  |-  2  e.  ZZ
589nn0zd 10760 . . . . . . . 8  |-  ( ph  ->  G  e.  ZZ )
5911nn0zd 10760 . . . . . . . 8  |-  ( ph  ->  E  e.  ZZ )
6058, 59zsubcld 10767 . . . . . . 7  |-  ( ph  ->  ( G  -  E
)  e.  ZZ )
61 dvdsaddr 13587 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  ( G  -  E
)  e.  ZZ )  ->  ( 2  ||  ( G  -  E
)  <->  2  ||  (
( G  -  E
)  +  2 ) ) )
6257, 60, 61sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  ||  ( G  -  E )  <->  2 
||  ( ( G  -  E )  +  2 ) ) )
6353, 62mtbid 300 . . . . 5  |-  ( ph  ->  -.  2  ||  (
( G  -  E
)  +  2 ) )
6463adantr 465 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( ( G  -  E )  +  2 ) )
65 2cnd 10409 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
668, 13, 65subaddd 9752 . . . . . . 7  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  2  <-> 
( ( G  -  E )  +  2 )  =  ( H  -  F ) ) )
6766biimpa 484 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
( G  -  E
)  +  2 )  =  ( H  -  F ) )
6867breq2d 4319 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
2  ||  ( ( G  -  E )  +  2 )  <->  2  ||  ( H  -  F
) ) )
6968notbid 294 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( -.  2  ||  ( ( G  -  E )  +  2 )  <->  -.  2  ||  ( H  -  F
) ) )
7064, 69mpbid 210 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( H  -  F ) )
7156, 70, 51mpjaodan 784 . 2  |-  ( ph  ->  -.  2  ||  ( H  -  F )
)
7252, 71jca 532 1  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   {cpr 3894   class class class wbr 4307  (class class class)co 6106   RRcr 9296   0cc0 9297    + caddc 9300    < clt 9433    - cmin 9610   2c2 10386   NN0cn0 10594   ZZcz 10661    || cdivides 13550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-n0 10595  df-z 10662  df-dvds 13551
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator